Ba space

{{Short description|Class of Banach spaces}} {{Lowercase title}} {{Use shortened footnotes|date=May 2021}} In mathematics, the '''ba space''' <math>ba(\Sigma)</math> of an algebra of sets <math>\Sigma</math> is the Banach space consisting of all bounded and finitely additive signed measures on <math>\Sigma</math>. The norm is defined as the variation, that is <math>\|\nu\|=|\nu|(X).</math>{{sfn|Dunford|Schwartz|1958|loc=IV.2.15}}

If Σ is a sigma-algebra, then the space <math>ca(\Sigma)</math> is defined as the subset of <math>ba(\Sigma)</math> consisting of countably additive measures.{{sfn|Dunford|Schwartz|1958|loc=IV.2.16}} The notation ''ba'' is a mnemonic for ''bounded additive'' and ''ca'' is short for ''countably additive''.

If ''X'' is a topological space, and Σ is the sigma-algebra of Borel sets in ''X'', then <math>rca(X)</math> is the subspace of <math>ca(\Sigma)</math> consisting of all regular Borel measures on ''X''.{{sfn|Dunford|Schwartz|1958|loc=IV.2.17}}

== Properties == All three spaces are complete (they are Banach spaces) with respect to the same norm defined by the total variation, and thus <math>ca(\Sigma)</math> is a closed subset of <math>ba(\Sigma)</math>, and <math>rca(X)</math> is a closed set of <math>ca(\Sigma)</math> for Σ the algebra of Borel sets on ''X''. The space of simple functions on <math>\Sigma</math> is dense in <math>ba(\Sigma)</math>.

The ba space of the power set of the natural numbers, ''ba''(2'''N'''), is often denoted as simply <math>ba</math> and is isomorphic to the dual space of the ℓ∞ space.

=== Dual of B(Σ) === Let B(Σ) be the space of bounded Σ-measurable functions, equipped with the uniform norm. Then ''ba''(Σ) = B(Σ)* is the continuous dual space of B(Σ). This is due to Hildebrandt{{r|Hildebrandt1934}} and Fichtenholtz & Kantorovich.{{r|FichtenholtzKantorovich1934}} This is a kind of Riesz representation theorem which allows for a measure to be represented as a linear functional on measurable functions. In particular, this isomorphism allows one to ''define'' the integral with respect to a finitely additive measure (note that the usual Lebesgue integral requires ''countable'' additivity). This is due to Dunford & Schwartz,{{sfn|Dunford|Schwartz|1958}} and is often used to define the integral with respect to vector measures,{{r|DiestelUhl1977_ChptI}} and especially vector-valued Radon measures.

The topological duality ''ba''(Σ) = B(Σ)* is easy to see. There is an obvious ''algebraic'' duality between the vector space of ''all'' finitely additive measures σ on Σ and the vector space of simple functions (<math>\mu(A)=\zeta\left(1_A\right)</math>). It is easy to check that the linear form induced by σ is continuous in the sup-norm if σ is bounded, and the result follows since a linear form on the dense subspace of simple functions extends to an element of B(Σ)* if it is continuous in the sup-norm.

=== Dual of ''L''∞(''μ'') ===

If Σ is a sigma-algebra and ''μ'' is a sigma-additive positive measure on Σ then the Lp space ''L''∞(''μ'') endowed with the essential supremum norm is by definition the quotient space of B(Σ) by the closed subspace of bounded ''μ''-null functions: :<math>N_\mu:=\{f\in B(\Sigma) : f = 0 \ \mu\text{-almost everywhere} \}.</math> The dual Banach space ''L''∞(''μ'')* is thus isomorphic to :<math>N_\mu^\perp=\{\sigma\in ba(\Sigma) : \mu(A)=0\Rightarrow \sigma(A)= 0 \text{ for any }A\in\Sigma\},</math> i.e. the space of finitely additive signed measures on ''Σ'' that are absolutely continuous with respect to ''μ'' (''μ''-a.c. for short).

When the measure space is furthermore sigma-finite then ''L''∞(''μ'') is in turn dual to ''L''1(''μ''), which by the Radon–Nikodym theorem is identified with the set of all countably additive ''μ''-a.c. measures. In other words, the inclusion in the bidual :<math>L^1(\mu)\subset L^1(\mu)^{**}=L^{\infty}(\mu)^*</math> is isomorphic to the inclusion of the space of countably additive ''μ''-a.c. bounded measures inside the space of all finitely additive ''μ''-a.c. bounded measures.

== See also == * List of Banach spaces

== References == * {{cite book |last1=Dunford |first1=N. |last2=Schwartz |first2=J.T. |date=1958 |title=Linear operators, Part I |publisher=Wiley-Interscience }}

<ref name=Hildebrandt1934>{{cite journal |last=Hildebrandt |first=T.H. |date=1934 |title=On bounded functional operations |journal=Transactions of the American Mathematical Society |volume=36 |issue=4 |pages=868–875 |doi=10.2307/1989829 |jstor=1989829 |doi-access=free}}</ref>

<ref name=FichtenholtzKantorovich1934>{{cite journal |last1=Fichtenholz |first1=G. |last2=Kantorovich |first2=L.V. |date=1934 |title=Sur les opérations linéaires dans l'espace des fonctions bornées |journal=Studia Mathematica |volume=5 |pages=69–98 |doi=10.4064/sm-5-1-69-98 |doi-access=free}}</ref>

<ref name=DiestelUhl1977_ChptI>{{cite book |last1=Diestel |first1=J. |last2=Uhl |first2=J.J. |date=1977 |title=Vector measures |series=Mathematical Surveys |volume=15 |publisher=American Mathematical Society |at=Chapter I}}</ref>

</references>

==Further reading== * {{cite book |last=Diestel |first=Joseph |date=1984 |title=Sequences and series in Banach spaces |publisher=Springer-Verlag |isbn=0-387-90859-5 |oclc=9556781 |url-access=registration |url=https://archive.org/details/sequencesseriesi0000dies}} * {{cite journal |last1=Yosida |first1=K. |last2=Hewitt |first2=E. |date=1952 |title=Finitely additive measures |journal=Transactions of the American Mathematical Society |volume=72 |issue=1 |pages=46–66 |doi=10.2307/1990654 |jstor=1990654 |doi-access=free}} * {{cite book |last1=Kantorovitch |first1=Leonid V. |last2=Akilov |first2=Gleb P. |title=Functional Analysis |date=1982 |publisher=Pergamon |isbn=978-0-08-023036-8 |doi=10.1016/C2013-0-03044-7}}

Category:Measure theory Category:Banach spaces